Smoothing effects for some derivative nonlinear Schrödinger equations without smallness condition
Smoothing effects for some derivative nonlinear Schrödinger equations without smallness condition
In this paper we study a smoothing property of solutions to the Cauchy problem for the nonlinear Schrödinger equations of type : A iut+uxx=N(u,u¯,uxux¯), t∈ℝ,x∈ℝ;u(x,0)=u0(x), x∈ℝ, where N(u,u¯,ux,ux¯)=K1|u|2u+K2|u|2ux+K3u2ux¯+K4|ux|2u+K5u¯ux2+K6|ux|2ux, the functions Kj=Kj(|u|2) satisfy Kj∈C∞([0,+∞);ℂ). This equation has been derived from physics. For example if the nonlinear term is N(u)=u¯ux21+|u|2 then equation …